A Noble-Gas-Centered Coordinate for Within-Period Atomic Property Trends

Abstract

We introduce a single dimensionless landscape function $J_{\mathrm{chem}}(\rho )=\cosh (\rho \ln \phi )-1$, $\phi =(1+\sqrt{5})/2$, on the noble-gas-centred coordinate $\rho =d/L_p\in [0,1)$, and show that it organizes four central atomic observables: first ionization energy \IE${}_1$, electron affinity EA, Mulliken electronegativity ${\chi }_M$, and Pearson chemical hardness $\eta$, on one periodic-table axis. The outward step $\Delta J_{\mathrm{chem}}^{+}$ delivers IE${}_1$; the inward gap $\Delta J_{\mathrm{chem}}^{-}=J_{\mathrm{chem}}(1)-J_{\mathrm{chem}}(\rho )$ delivers EA and $\eta$; ${\chi }_M$ follows by Mulliken's identity. Three results establish the empirical content. (i) The within-period IE${}_1$ envelope reproduces the full noble-gas-to-alkali ordering across periods 2--6: of 34 atoms compiled across periods 2-4, 26 lie on the predicted monotone descent and the 8 upward deviations occur exactly at the textbook anomaly sites $\{p^3,d^5,f^7,s^2,d^{10}\}$. (ii) Two golden-ratio identities, ${\mathrm{IE}}_1(G_p)/{\mathrm{IE}}_1(G_{p+1})\approx {\phi }^{1/4}$ on three heavy noble-gas pairs and ${\mathrm{IE}}_1(\text{halogen})/{\mathrm{IE}}_1(\text{alkali})\approx {\phi }^2$ on four within-period pairs, agree with NIST data to MAD $\approx 1\%$ and $\approx 5\%$, respectively. (iii) The shared kernel $\Delta J_{\mathrm{chem}}^{-}$ provides single-parameter analytical fits to EA across periods 4--6 (MAE $0.3$--$0.4$~eV), to Pearson hardness $\eta$ across periods 2--4 (MAE $\sim 1$~eV on noble-gas maxima up to $10.8$~eV), and to Mulliken ${\chi }_M$ across a 15-atom four-class benchmark ($R^2=0.73$). \edit{At the period-averaged scale level, the shared-kernel relation $\mathrm{EA}/\eta \approx C_{\mathrm{EA}}^{(p)}/C_{\eta }^{(p)}$ is supported on period-4 NIST data: the empirical nine-atom mean $\overline{\mathrm{EA}/\eta }=0.180$ agrees with the predicted constant $0.182$ to better than $1\%$, although individual-atom scatter ($\sigma \approx 0.13$) is much larger.

32 pages, 8 figures